I haven't filed Federal yet but have filed my Louisiana State* return as it is very simple.
I sent my return via registered post from England, tracked it and saw that it had been delivered on March 20. I received a refund check in the mail here in England on April 27, which is about exactly the timeline I was used to while living in Texas.
*since retiring in 2010 I left Louisiana for Texas and have been non-resident in Louisiana since then. However I do receive a non-qualified pension reported by W2 from my ex-employer in Louisiana, and that requires me paying State taxes on that income. I got my ex-employer to withhold the minimum but I still end up claiming a refund of about $350 every year and if you file by post they won't do a direct bank deposit, hence the paper check. I still have a US bank account and do a mobile check deposit so for me it's not so bad.